Proof of Theorem 4. 
To analyze the large-\(a_{2m+1}\)  behavior of the rogue wave \(u_N(x,t)\)  in the neighborhood of the origin, where \(\sqrt{x^2+t^2}=O(1)\) , we first rewrite the \(\sigma _n\)  determinant (REF ) into a \(3N\times 3N\)  determinant [1]}
\( \sigma _{n}=\left|\begin{array}{cc}\textbf {O}_{N\times N} & \Phi _{N\times 2N} \\-\Psi _{2N\times N} & \textbf {I}_{2N\times 2N} \end{array}\right|,\) 
